The Hindustan Republican Association (HRA) was founded with the purpose of encouraging Indian youth to embrace socialism. Option 4 is the correct answer.

Option 1, to set up a republican government in Hindustan, is not the goal of HRA. The organization focused on promoting socialism rather than advocating for a specific form of government.

Option 2, to organize armed rebellion in India, is also not the main objective of HRA. While the organization did engage in some acts of violence against British colonial rule, their primary aim was to promote socialist ideals rather than solely focusing on armed resistance.

Option 3, to persuade Indian people to participate in elections, is not in line with the goals of HRA. The organization aimed to instill a sense of socialist ideology and activism among Indian youth, rather than focusing on electoral politics.

In summary, the Hindustan Republican Association was founded to encourage Indian youth to embrace socialism and did not prioritize the establishment of a republican government, armed rebellion, or participation in elections.